Emergent Non-Markovianity in Logical Qubit Dynamics

TL;DR

This work shows that logical qubits encoded by quantum error-correcting codes can exhibit non-Markovian dynamics even when the underlying physical noise is Markovian, due to memory stored in syndrome qubits when QEC does not return data to the codespace after every round. The authors define button-theoretic Markovianity and a gadget retraction formalism to map physical processes to effective logical dynamics, proving that gate composability can fail in realistic two-round syndrome-extraction scenarios and that non-Markovianity is generically present for stabilized codes with $d\ge3$. They illustrate these effects with the 3-qubit repetition code and the [[5,1,3]] code, and connect the phenomena to open-system perspectives and Markovian vs non-Markovian characterizations of complex, memory-bearing environments. The results have practical implications for interpreting logical QCVV protocols like GST and RB and for designing fault-tolerant circuits that minimize emergent non-Markovian effects in early devices.

Abstract

Logical qubits encoded in quantum error correcting codes can exhibit non-Markovian dynamical evolution, even when the underlying physical noise is Markovian. To understand this emergent non-Markovianity, we define a Markovianity condition appropriate to logical gate operations, and study it by relating logical operations to their physical implementation (operations on the data qubits into which the logical qubit is encoded). We apply our analysis to small quantum codes, and show that they exhibit non-Markovian dynamics even for very simple physical noise models. We show that non-Markovianity can emerge from Markovian physical operations if (and only if) the physical qubits are not necessarily returned to the code subspace after every round of QEC. In this situation, the syndrome qubits can act as a memory, mediating time correlations and enabling violation of the Markov condition. We quantify the emergent non-Markovianity in simple examples, and propose sufficient conditions for reliable use of gate-based characterization techniques like gate set tomography in early fault-tolerant quantum devices.

Figures (7)

• Figure 1: We conceptualize a qubit (physical or logical) or a multi-qubit quantum register as a black box. Logic operations such as initialization, gates, and measurements correspond to buttons that can be pushed in sequence to execute quantum circuits. Using this paradigm, we define button-theoretic Markovianity to hold if and only if the quantum register experiences the same dynamical transformation every time a particular button is pressed. Protocols for characterizing physical qubits, such as gate set tomography, typically assume button-theoretic Markovianity. We say that an error-corrected register satisfies logical Markovianity if button-theoretic Markovianity holds for the set of all logical operations, including the logical idle.
• Figure 2: Two rounds of syndrome extraction in the 3-qubit repetition code with bitflip errors on the ancilla qubits. The bitflip in the first round leads to an incorrect readout of the syndrome, causing the decoder pick the wrong correction. The result is a single qubit error on the data qubits that won't change the outcome of an error-corrected logical measurement. In the second round, an ancilla error causes a second bitflip on the data qubits, leading to a combined error that will flip the outcome of an error-corrected logical Z measurement.
• Figure 3: (\ref{['fig:surv']}) Polarization in the $[3,1,3]$ code and the $[[5,1,3]]$ code under repeated rounds of noisy syndrome extraction. A constant exponential decay is expected for a Markovian logical process. In log scale, non-Markovianity is demonstrated by a deviation of the decay from a straight line. The decay curve becomes quickly indistinguishable from straight after just a few rounds. This indicates logical processes where most of the non-Markovianity is relegated to the first few rounds. (\ref{['fig:decay']}) The logical error rate per round of the $[3,1,3]$ code and the $[[5,1,3]]$ code. The absolute value of the change in error rate is plotted in (\ref{['fig:decaychange']}). It is clear from examining the raw data that the error rate change falls of exponentially in the number of rounds until the changes fall below the machine precision ($\sim 10^{-15}$). In summary, the data indicate logical processes where most of the non-Markovianity is relegated to the first few rounds, yet persists indefinitely.
• Figure 4: A graph representing the action of 1 noisy QEC round consisting of syndrome extraction and correction in the 3-qubit code. The nodes correspond to computational basis states and the arrows correspond to the action of the noisy recovery map on the states given a particular error, each of which is denoted by a color (e.g. blue corresponds to a bitflip on the first ancilla qubit during syndrome extraction, the blue arrows representing the map $\widetilde{\mathcal{R}}_{10}$). In the absence of syndrome extraction error (the black arrows), the action of the QEC round is to "cool" the system to the codespace, but syndrome errors will cause the experimenter to confuse one syndrome space for another. A QEC round with syndrome extraction error $(i,j)$ will cool the system to the subspace with syndrome $(i,j)$ via the map $\widetilde{\mathcal{R}}_{ij}$. This leads to logical transition depending on the syndrome of the input state.
• Figure 5: The encoding unitary for the 3-qubit repetition code. The encoding unitary, denoted $U_E$ relates states of the subsystems of syndrome qubits and logical qubits to an encoded state of data qubits. The green, blue, and purple wires correspond to the syndrome qubits, logical qubits and data qubits respectively. When the syndrome qubits are in the all-zeroes state, the corresponding data qubit states are in the codespace, (i.e. for the 3-qubit repetition code, and an arbitrary logical qubit state, $\ket{\psi}$, $U_E\ket{00}\otimes\ket{\psi}=\ket{\Bar{\psi}}$. For a non-trivial syndrome ${s}$, $U_E$ maps the syndrome qubit state $\ket{{s}}$ to a subspace determined by a choice of Pauli correction, $R({s})$, such that for an arbitrary logical qubit state $\ket{\psi}$, $U_E\ket{{s}}\otimes\ket{\psi}=R^{\dagger}({s})\ket{\Bar{\psi}}$ (see Table \ref{['tab:encunitary']}).

Theorems & Definitions (14)

• Theorem 3.1
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• proof
• proof
• Definition 1: Button-theoretic Markovianity
• Definition 2: Gate Composability
• Proposition B.1